Questions: The graph below represents the solution set of which inequality? x^2 - 2x - 8 < 0 x^2 + 2x - 8 < 0 x^2 - 2x - 8 > 0 x^2 + 2x - 8 > 0

Solution Steps

The graph shows the solution set for the inequality. The solution set is the interval between -2 and 4, not including -2 and 4.

The roots of the quadratic inequalities are the points where the quadratic expression equals zero. We need to find which quadratic inequality has roots at -2 and 4.

1. \(x^2 - 2x - 8 < 0\)

   • Factor: \((x - 4)(x + 2) < 0\)
   • Roots: \(x = 4\) and \(x = -2\)
   • The inequality is satisfied between the roots, i.e., \(-2 < x < 4\).

2. \(x^2 + 2x - 8 < 0\)

   • Factor: \((x - 2)(x + 4) < 0\)
   • Roots: \(x = -4\) and \(x = 2\)
   • The inequality is satisfied between the roots, i.e., \(-4 < x < 2\).

3. \(x^2 - 2x - 8 > 0\)

   • Factor: \((x - 4)(x + 2) > 0\)
   • The inequality is satisfied outside the roots, i.e., \(x < -2\) or \(x > 4\).

4. \(x^2 + 2x - 8 > 0\)

   • Factor: \((x - 2)(x + 4) > 0\)
   • The inequality is satisfied outside the roots, i.e., \(x < -4\) or \(x > 2\).

Final Answer